A polarized view of string topology
A polarized view of string topology
复制标题
串拓扑的极化视图
DOI：
10.1017/cbo9780511526398.008
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
V. Godin
中科院分区：
文献类型：
--
作者：
R. Cohen;V. Godin
设M是一个闭连通流形，LM是它的回路空间。本文描述了h_*（LM）中的闭弦拓扑运算，其中h_* 是支持M的定向的广义同调理论.我们将证明这些运算使h_*（LM）具有无可数单位的交换Frobenius代数的结构。等价地，它们描述了一个与h_*（LM）有关的正边界二维拓扑量子场论.这意味着只要q >0，就存在对应于具有p个输入边界分量和q个输出边界分量的任何表面的操作。不存在计数是因为不存在与圆盘D^2相关联的运算，可以看作是从圆到空集的配边。我们将研究构造这种运算的同调障碍，并证明为了使这种运算存在，必须取h_*（LM）为与循环空间相关联的适当的同调pro-object。出于这个动机，我们引入了一个prospectrum与LM时，M有一个几乎复杂的结构。给定这样一个流形，它的环空间有它的切丛的正则极化，这是定义这个原谱所需要的基本特征。我们称之为LM的“极化Atiyah -对偶”。一个适当的同调理论适用于这个prospectrum将是一个候选的理论，支持弦拓扑操作与任何表面，包括封闭的表面。
Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h_*(LM), where h_* is a generalized homology theory that supports an orientation of M. We will show that these operations give h_*(LM) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h_*(LM). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q >0. The absence of a counit follows from the nonexistence of an operation associated to the disk, D^2, viewed as a cobordism from the circle to the empty set. We will study homological obstructions to constructing such an operation, and show that in order for such an operation to exist, one must take h_*(LM) to be an appropriate homological pro-object associated to the loop space. Motivated by this, we introduce a prospectrum associated to LM when M has an almost complex structure. Given such a manifold its loop space has a canonical polarization of its tangent bundle, which is the fundamental feature needed to define this prospectrum. We refer to this as the "polarized Atiyah - dual" of LM . An appropriate homology theory applied to this prospectrum would be a candidate for a theory that supports string topology operations associated to any surface, including closed surfaces.